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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2010

    added a simple proposition and proof to essential geometric morphism in a new section "Properties".

    Also replied to Mike's query box comment.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMar 10th 2010

    Re-replied and created locally connected geometric morphism.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2010
    • (edited Mar 10th 2010)

    Thanks, Mike.

    I did the following at essential geometric morphism:

    • turned the query box discussion into genuine text in the Idea-section;

    • added a section Definition-Refinements with the locally connected case and two other refinements of "essential";

    • added references for these cases.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2010
    • (edited May 6th 2010)

    A question:

    is there an established special name for the situation where we have an essential geometric morphism

    (f !f *f *):ET (f_! \dashv f^* \dashv f_*) : E \stackrel{\to}{\stackrel{\leftarrow}{\to}} T

    with the special property that f *f^* is full and faithful ?

    I know (from Johnstone and Lawvere’s article referenced at essential geometric morphism) that in the case that we have yet one more right adjoint f !f^!

    (f !f *f *f !) (f_! \dashv f^* \dashv f_* \dashv f^!)

    and if that f !f^! is full and faithful, then one says that ff is local ,

    But I am wondering about f *f^* being full and faithful. (Because, unless I am mixed up, this is the case for the terminal geometric morphisms out of the objects in the class of locally contractible (oo,1)-toposes that I know how to build).

    Probably this is a stupid question with an evident answer, but right now it escapes me.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 6th 2010

    Yes! It’s called a connected topos.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2010
    • (edited May 6th 2010)

    Yes! It’s called a connected topos.

    Ah, nice. Okay, so this proves publically that I stil haven’t read the Elephant the way I should, but at least it’s the perfect answer for my purpose. :-)

    Mike, I now want to make the following definition, but please give me a sanity check:

    of course I want to say now that a contractible (,1)(\infty,1) -topos is a locally contractible (infinity,1)-topos H\mathbf{H} such that LConst:GrpdHLConst : \infty Grpd \to \mathbf{H} is a full and faithful (infinity,1)-functor.

    Here is one consistency check that this makes sense: in the case that LConstLConst is full and faithful we have that

    • the unit Id GrpdΓLConstId_{\infty Grpd} \to \Gamma LConst is an equivalence .

    This implies that the shape of H\mathbf{H} in the sense of shape of an (infinity,1)-topos is that of the point. Which clearly matches the idea of H\mathbf{H} being “contractible” .

    I am inclined to make that into an nLab page connected (infinity,1)-topos.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2010
    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeMay 6th 2010

    Would you have any objection to having a separate page locally connected topos about the 1-dimensional version?

    I’m not sure I like calling that notion “contractible” – wouldn’t something like “\infty-connected” be closer to the mark? “Contractible” to me means “equivalent to a point,” which such a topos evidently is not always (or so I gather, otherwise you wouldn’t be interested in it). Is this at all similar to the notion of \infty-connected object in an (,1)(\infty,1)-topos (which also need not be trivial)?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2010
    • (edited May 7th 2010)

    Would you have any objection to having a separate page locally connected topos about the 1-dimensional version?

    Right, I was thinking about that, too. We should do that. But I won’t do anything else tonight, need to catch some sleep.

    I’m not sure I like calling that notion “contractible” – wouldn’t something like “∞-connected” be closer to the mark?

    Maybe you are right.

    “Contractible” to me means “equivalent to a point,” which such a topos evidently is not always (or so I gather, otherwise you wouldn’t be interested in it).

    Right, so this touches on the crucial interesting point here: these “\infty-connected” \infty-toposes are in a way fat points with structure. For instance with smooth structure.

    Consider the underlying site: the objects of CartSp are all contractible spaces. It is helpful to think of them (up to diffeomorphism) as the open nn-balls. An nn-ball is just a fat point, topologically. But crucially here the fact that maps are smooth maps remembers the smooth structure. So an \infty-groupoid modeled on the smooth nn-balls, i.e. an oo-stack on CartSpCartSp is much like a bare Kan complex, the only difference being that for around every point in the k-cells of the Kan complex, I have the information of what the ways are to extend that point smoothly to an open-ball-shaped family of points in its neightbourhood.

    So Sh (,1)(CartSp)Sh_{(\infty,1)}(CartSp) differs from Sh (,1)(pt)Sh_{(\infty,1)}(pt) just a little bit, but by a crucial bit. This is I think what the abstract formalism is seeing: that Sh (,1)(CartSp)Sh_{(\infty,1)}(CartSp) is “locally \infty-connected and \infty-connected” is a reflection of the fact that all the objects of CartSpCartSp are.

    Is this at all similar to the notion of ∞-connected object in an (∞,1)-topos (which also need not be trivial)?

    I was wondering about that today. One should look at the (,1)(\infty,1)-category of (,1)(\infty,1)-toposes and see if these conditions on the terminal morphism induce corresponding properties as connected objects. It’s probably an easy argument along the lines: if we have an essential geometric morphism to the terminal oo-topos with the inverse image full and faithful, then this means in the (,1)(\infty,1)-category of (,1)(\infty,1)-toposes we have a exhibited the terminal object as a retract of our (,1)(\infty,1)-topos.

    I’ll think about it. But not tonight. I need to call it quits now.

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